Abstract

This paper describes some properties of the eigenvalue equation $\psi_{n+1} + \psi_{n-1} + 2\alpha \cos (2\pi\beta n + \Delta)\psi_n = E\psi_n$. This is an example of the more general problem of a Hermitian eigen value equation in the form of a difference equation with periodic coefficients. These equations arise in solid state physics; they occur in connection with tight-binding models for electrons in one-dimensional solids with an incommensurate modulation of the structure, and in models for the energy bands of Bloch electrons moving in a plane with a perpendicular magnetic field. The model studied has a critical point when $\alpha$ = 1. Following some earlier work by Azbel (Azbel, M. Ya., Phys. Rev. Lett. 43, 1954 (1979)), an approximate renormalization group transformation is derived. This predicts that the spectrum and eigenstates have a remarkable recursive structure at the critical point, which is dependent on the expansion of $\beta$ as a continued fraction. Also, when $\beta$ is an irrational number, there is a localization transition from extended states to localized states as $\alpha$ increases through the critical point. This localization transition, which was previously discovered by Aubry & Andre (Aubry, S. & Andre, G. Ann. Israel phys. Soc. 3, 133 (1979)) using the Thouless formula for the localization length, is explained by the renormalization group transformation derived here.